

ElectroMagnetic "Physics"
This section reviews basic elements of electromagnetic propagation, from the speed of light to choice of transmission wavelength and the formation of directional transmission lobes. It touches on matters of refraction and diffraction, on polarization, and on the phenomena collectively known as Faraday effects. It is intended only as an introduction to these topics.
I. Wavelength and Radio Frequency
II. Time and Distance
III. Polarization, Propagation and Attenuation
IV. Size Matters
Natural Laws Governing Propagation of Electromagnetic Waves
A detailed appreciation of the MNR domain requires more than a passing familiarity with the natural phenomena that determine the characteristics and behavior of the many systems in the domain. The following paragraphs, which describe some of what might be considered "natural laws of physics" and their practical influences on the design and characterization of radar systems, are presented here only as a guide to further research. The focus is deliberately on the MNR domain, although the effects may be considered applicable to radar in general.
Wavelength and Radio Frequency
These two terms are used interchangeably in the radar engineering community, which is not surprising. They are truly inseparable, being bound by a single factor: the celerity of light. This natural behavior was first measured somewhat inaccurately by Olaus Roemer, of Denmark, in 1676. James Bradley, of England, obtained an incredibly precise measurement of 301,000 km/second in 1728, by observing stellar aberrations. In France, 135 years later, Leon Foucault conducted an experiment involving spinning mirrors and deduced a speed of 298,000 km/second. Nowadays, we know it even more precisely: 299,792,458 meters/second in a vacuum; but it is rarely used with this much precision, since we don't exist in a vacuum. It is much more common practice to use a rounded value of 300,000 km/second, a celerity straddled almost equally by those distant physicists' calculations. We can only wonder at their experimental skills.
The correlation of wavelength and frequency is simple: one is a physical unit of length, usually measured in scaled meters, and the other is the number of those physical units in a 300,000 km (3*10^{8} m) span. It is usual to use the Greek letter 'λ' to represent wavelength, the letter 'c' to represent the celerity of light and 'F' to represent frequency; thus, the inseparable nature of frequency and wavelength:
F * λ = c
From this, we can express a radio frequency of 3050 MHz (3050 *106 Hz) as:
λ = 3*10^{8} : 3050 *10^{6} meters, or 9.836 cm
Practical Considerations: Doppler and Sidereal Effects
Christian Andreas Doppler published his findings 'On the Colored Light of the Double Stars and Certain Other Stars of the Heavens' to the Royal Bohemian Society of Sciences in 1842, long after he had proven the existence of frequency shifts induced by motion with a wellknown experiment involving trains and trumpeters. What we know today as the Doppler Effect is a reflection of what he demonstrated: expansion of waves proportional to their vector away from an observer, compression proportional to the vector towards an observer. In any circumstance where a motion vector is involved, there is a difference between radiated wavelength and observed wavelength that is directly related to the relative motion vector. That difference is most easily expressed as a frequency shift, and it may be calculated as the number of radiated wavelengths that may be divided into the displacement caused by the motion vector. To illustrate, the motion vector of an aircraft traveling at a speed of 750 km/hour, or 200 m/S, and transmitting a radio signal with a wavelength of 0.03 m, will cause a maximum frequency shift, F_{Shift}, of 200 : 0.03, or 6667 Hz. There are several key points to note:
 The maximum frequency shift occurs only along the axis of motion, either towards, when the shift results in an increase in apparent RF; or away, when the shift results in a decrease.^{ }
 An observer not on the axis of motion measures a smaller shift that is dependent on the deflection angle, the angle subtended between the axis of motion of the signal source and the location of the observer. In trigonometric terms, when observer and source are coplanar, the scaling factor is the cosine of this deflection angle. When they are not, the scalar factor is the product of the angle of deflection and the angle of depression, the vertical angle subtended between the axis of motion of the signal source and the location of the observer.
^{}
 When the observer is also the signal source, as is commonly the case in radar, the effects are doubled. This is because the wavelength reflected to the radar has been shifted by the motion of the transmitting radar, and this reflection is influenced in turn by the receiver's motion. ^{ }
Sidereal effects may be viewed as somewhat orthogonal to Doppler effects: the observed Doppler shift is invariably at its greatest when the observer is on the axis of motion of a source, whereas the sidereal effect is at its greatest when the location of the observer lies at 90° to the axis of motion. Ordinarily, the term 'sidereal time' is used in astronomy, and it refers to a fixed point in the celestial sphere. In radar terms, especially for the observer of radars, 'sidereal time' may be viewed as differences between actual and observed behaviors, particularly those related to antenna scan arising from displacement. We may express the rotational behavior of an MNR antenna either as a rate ('24 rotations per minute'), or as a periodicity ('2.5 seconds per rotation'). Accurate measurement of that behavior may be complicated when the observation scenario involves any element of dynamism.
Consider two highspeed craft following reciprocal tracks that will result in them passing at close quarters:
 At the horizon or beyond, an observer measuring the interval between successive sweeps of the other vessel's radar will see little or no difference, sweeptosweep.
 As the vessels close, the observer will see progressive alterations in the interval between successive sweeps. This is because, unless the vessels are on a collision course, their relative aspects are changing, and the radar antenna may have to turn more or less than 360° between successive illuminations.
 At their closest point of approach, when any Doppler effect will be at its very lowest, the aspectchange is at its most dramatic. The more fleet the vessels and the more ponderous the turn of the radar antenna, the greater the change in aspect between successive sweeps of the radar, even though it has continued to turn through 360° at a constant rate.
 If the observer's craft is airborne, it may directly overfly the radar. In this circumstance, the nearinstantaneous aspect change between illuminations may be as great as 180°, depending on differences in height between antenna and observer.
The sidereal effect, then, arises when measurements are not taken with a geometricallyfixed relationship. It is not an 'error' per se, simply a bias that must be considered when determining the accuracy of any measurement. Ideally, rotation should be characterized at the maximum possible distance, where sidereal effects are minimized; otherwise, it is necessary to know the geometry, the motion vectors and the direction of rotation of the radar antenna in order to arrive at an accurate estimate of its periodicity. Even so, characterization may be complicated: considering our two highspeed craft, it is highly likely that a closequarters situation would stimulate an alteration of course; unless the antenna drivesystem employs compensatory measures to damp the effect of vessel maneuvers, further perturbation of observed behavior may occur.
Time and Distance
Just as with antenna rotation, we may express the pulsing rate of a radar system either as the average number of pulses that occur in a period, or as the average interval between pulses. Manufacturer literature tends to express both antenna rotation and pulsing behavior by using a "rate" measurement. Outside that literature, it is much more common to refer to MNR antenna scans by periodicity than by rate, whereas pulse behavior may be described equally by periodicity and by rate. Conversion between the two is even simpler than between frequency and wavelength: Periodicity is usually expressed either as microseconds (μS) or as milliseconds (mS) between pulses; frequency, as the number of pulses that occur in 1 second, typically with units of Hz or kHz. Ordinarily, in the community engaged in characterizing radar behavior, terms such as PRI (pulse repetition interval) and PRF (pulse repetition frequency) are used; these may be either specific values, or indicators of the average behavior, and it should be considered the reporter's responsibility to assure clarity.
As discussed above, the celerity of light, and of all electromagnetic waves, is approximately 300,000 km/S, or 300 m/μS. In 1000 μS, then, a pulse of electromagnetic energy would travel a total distance of 300,000 m, or 300 km. For the classical radar architecture, that total distance involves the flight time from the radar to a target and back again, and so a target's range would be measured as 150 km. Rationally, if a constant PRI of 1000 μS is used, then the radar is said to have a maximum unambiguous range of 150 km, which is equal to 91 statute miles, or 81 nautical miles. The term 'unambiguous' is used to discriminate this range from the maximum detection range, the maximum distance that a pulse may fly out and then back to the radar with sufficient energy that it may be perceptible to the radar. Radar designers may use a variety of techniques to extend the maximum unambiguous range, especially in military radars. These same techniques may be employed in MNRs, but rarely if ever to increase the maximum unambiguous range: the same techniques may be used to reduce mutual interference, a cruciallyimportant issue as the seaways of the world become ever more crowded. Some of the techniques, those that are in common use by MNR designers, are described here.
Since the topic of unambiguous range has been raised, it worth digressing to discuss 'horizon' range, the distance at which an object may be detected by a sensor. Since Spaceship Earth is notionally spherical, the maximum distance at which a small object on its surface may be seen is directly related to two things: the height of the observer, and the radius of the sphere. This maximum distance, d, in kilometers, of the horizon is defined by the expression:
d = √ (2Rh + h^{2}),
where R is the radius of the Earth, approximately 6371 km, and h is the height of the observer, also expressed in km. This method is quite precise, even for satellites, but perhaps rarely used in radar considerations; a more convenient form, which is slightly less accurate but still quite useful when h is very small compared to the Earth's radius, as with MNR systems, requires only the height of the observer, in meters:
d = √ (13h)
For the purposes of discussing distance as it relates to pulse transit times, where time is expressed in μS and the celerity is expressed in metric form, the simplified version is used here. It shows that, for a radar height of 10m, the horizon is at a distance of 11.40km; thus, a pulse of energy would require 76 μS to travel out to the true horizon and back.
If a radar were to transmit a pulse every 76 μS then it would have an average PRF of 13,158 Hz  more than three times greater than the very highest PRF published for any of the MNRs tabulated in our Case Studies. Clearly, the pulse continues on beyond the true horizon; after grazing the far horizon it continues onward, and could reflect off a 10m high object some 11.40km beyond the horizon and still be able to return to the point of origin. If the reflective object is lower than 10m, it must be nearer to that grazingpoint; if it is higher, then it can be further away. So, the detection range is dependent on both the height of the observer and the height of the reflector:
d = √ (13h_{1}) + √ (13h_{2}), or d ≈ 3.6 * (√ h_{1} + √ h_{2})
Useful though these approximations are, they do not tell the full story, as they ignore entirely the effects of refraction, i.e. the influence of the Earth's atmosphere on the propagation of electromagnetic waves. The practiced navigator will routinely check the performance of his ship's gyrocompass by measuring the azimuth bearing of the setting sun at the moment when its full diameter is barely above the visible horizon, and checking that bearing against astronomical predictions; at that instant, sunset, the sun is truly fully below the horizon, even though its entire span may be visible; refraction at work. Put another way, the visible horizon may be considerably beyond the physical horizon, because of refraction caused by the atmosphere, the extent of which is quite variable. When dealing with the refraction of visible light, it is usual to compensate by using a virtual radius for the Earth which is approximately onefifth greater than its true radius. To understand the effect of refraction at microwave frequencies, with very much longer wavelengths than visible light, requires a brief detour to consider polarization and propagation.
Polarization and Propagation
Polarization
After a humdrum sevenyear apprenticeship as a bookbinder and bookseller, Michael Faraday (17911862) became one of Britain's leading experimental scientists, with notable achievements in chemistry, magnetism and electricity. An early experiment in chemistry led him to a 'heavy glass' compound, and that in turn led to possibly his most notable achievement: the discovery, in 1845, that the behavior of light passing through a glass prism might be altered by the exertion of magnetic force ^{[}^{1}^{]}. What he termed diamagnetism is now known as the Faraday Effect, a discovery that established conclusively the relationship between visible light and magnetism; nowadays, we know that the Faraday Effect influences all electromagnetic wave energy, and that its principles underlie the behavior of the magnetron. Today, we also know that an electromagnetic wave comprises two components, each of which varies in magnitude simultaneously and sinusoidally  an electrical wave and a magnetic wave; and that these components vary perpendicularly both to each other and to the direction of propagation. At their simplest, antennas radiate a linearlypolarized plane wave, and it is usual in RF engineering to reference the orientation of this plane to the electrical field. Thus, a simple radiator  a dipole; or an aperture in a leakywaveguide antenna  will radiate a vertically or horizontallypolarized linear wave. Other configurations can be used to create a slantpolarized wave or even a continuouslyrotating ('circular') polarization; however, since these are not used in MNR antenna design, they are not discussed further here. For the MNR antenna, it is important to note several features:

Antennas enjoy a characteristic known as reciprocity: they are generally good only at receiving waveforms that they are good at radiating. Thus, a horizontallypolarized slottedwaveguide antenna, with its vertical slots, will be wellmatched to horizontallypolarized waveforms  and poorlymatched to verticallypolarized waveforms.

Both the linearity and the plane of polarization may be substantially modified by Faraday Effects that vary with wavelength, pathlength and other factors beyond the scope of this introduction. These effects may be negligible at wavelengths shorter than approximately 6 cm, but may cause some rotation of the polarization plane at 10 cm wavelengths, potentially degrading the reciprocity of a linearlypolarized antenna structure.

The closer an antenna to a surface, the more that its radiations interact with it. In the case of the MNR antenna, which is often less than 25m from the sea surface, electricalplane interaction with the sea can result in considerable attenuation and hence loss of range; thus, it is not only convenient to manufacture MNR antennas with horizontal polarization, it is also important to their performance.
Propagation
Paraphrasing Faraday's discovery, electromagnetic waves may be influenced substantially by the exertion of magnetic pressure. This has important consequences for RF propagation, since the Earth's ionosphere exerts continuous and continuouslyvarying extreme magnetic pressure. For the purpose of this handbook, it is necessary to simplify, by asserting that this magnetic pressure causes distortions in both the plane and the linearity of a planepolarized electromagnetic wave, and contributes to refraction  the change in direction of a wave due to a change in velocity that occurs as it passes from one medium to another.
The impact of the ionosphere, which varies according to the position of the sun and the incidence of solar storms, is one of several phenomena that cause electromagnetic waves to curve in their trajectory, usually downward towards the Earth's surface. Another, even more dominant influence, is the troposphere, the turbulent lower level of the atmosphere where most weather phenomena are visible. This layer varies in depth, from approximately 16 km in the tropics to around half that value at the poles, and comprises around threequarters of the atmosphere's mass, including virtually all of its watervapor content. The refractive index of the troposphere varies inversely with elevation, and the celerity of electromagnetic propagation varies inversely with the refractive index; thus, as elevation increases, so does celerity; and this differential behavior causes 'bending' of the energy back towards the more dense lower elevations. Because of its turbulent nature, the effect of the troposphere is rather less predictable than that of the ionosphere. Nonetheless, a general characteristic may be asserted: that, under normal conditions where temperature and density decline with elevation, the troposphere exerts a downward pressure, causing 'lineofsight' propagation to follow a curved trajectory that may approximate the curvature of the Earth and effect substantial increases in the usable horizon range. This normal behavior is more marked at longer wavelengths than at shorter wavelengths; thus, 10cm radars may benefit more than 3cm radars from its influence.
Refraction may have many modes, and there are also many other effects that influence the path of electromagnetic propagation. Diffraction, for instance, causes electromagnetic energy to be detectable even in socalled shadow zones; and ducting, one of the principal causes of mirages such as the Novaya Zemlya effect wellknown to Arctic navigators, may have a dramatic influence on propagation. These effects are well beyond the scope of a short text on MNR systems, and a simple assertion must suffice: that, generally, refraction increases with wavelength, varies somewhat with meteorological conditions and is influenced by the Earth's magnetosphere.
Because of the many independent variables, it is possible to derive only an approximation for radar's effective horizon range. It is common practice to use an approach somewhat like that used for estimating visible horizon (where the Earth's radius is assumed to be onefifth greater than its true radius), using a somewhat greater 'correction factor.' For RF propagation, the Earth is assumed to have a radius onethird greater than its true value of 6371 km, i.e. 8495 km. From this, an approximation of radar horizon range may be derived:
Radar Horizon Range (km) ≈ 4.12 * (√ h_{1} + √ h_{2}),
where h_{1} and h_{2} are the heights of the radar and the target respectively, in meters.
Attenuation
One of the more significant variables that affect radar horizon range is the water content of the atmosphere, which may vary considerably with temperature and can introduce significant attenuation to the propagation of electromagnetic waves. Attenuation due to water vapor content^{[2]} varies somewhat inversely with wavelength; however, at wavelengths shorter than about 3 cm its impact accelerates nonlinearly, and indeed there are a number of shorterwavelength regions which are entirely unusable for radar purposes. Full exploration of this is again beyond the scope of a text on MNRs; however, as a broad generalization, it is asserted that 10cm wavelengths are relatively unaffected by precipitation and water vapor when compared with 3cm wavelengths. Furthermore, it is asserted that longer wavelengths will generally penetrate cloud and rainfall better, and hence that targets will be less obscured by these natural phenomena than at shorter wavelengths.
Size Matters
Antennas
In all, it may seem  from the effects of refraction and water vapor  that it would be most preferable for an MNR to use a longer wavelength, to maximize the inatmosphere detection range and to minimize the performance degradation that is caused by rainfall. Be that as it may; there are nonetheless many more MNRs operating at 3cm wavelengths than at 10cm wavelengths, as is obvious from the various case studies. The reason for this is a direct correlation between antenna size and radiated wavelength: the longer the wavelength, the larger and heavier the antenna necessary to form a useful directional beam. Put another way, the more physical wavelengths in an antenna dimension, the narrower the beam that may be formed in that dimension, and the greater the proportion of the total energy that is radiated in the desired direction. Without delving into antenna theory, which is beyond the scope of this brief review, in the ideal antenna the relationship between dimension and beamwidth may be expressed as:
Dimension Beamwidth = λ : d radians, where d = dimension length in meters,
≡ 57.3 * λ : d degrees.
The perfect antenna doesn't exist, and in any case, for radar it may not be desirable. Consider the parabolic dish antenna: the perfect antenna would require uniform illumination of the entire surface, and thus of the edge, where there would be substantial spillover and where a phenomenon known as 'knifeedge' diffraction would cause very marked sidelobe structures, both of which degrade the antenna efficiency. To ameliorate this:
 The antenna designer may taper the illumination so that there is less at the perimeter of the dish; naturally, this results in reduced efficiency, in return for some reduction in spillover and diffraction.
 The designer may increase the diameter of the dish to create a narrower beamwidth; this results in a longer perimeter, and hence increased 'knifeedge' diffraction and sidelobes; contrarily, the narrower the beam, the more prominent the resultant sidelobes  natural effects that are not easily overcome.
 Another option is to reduce the wavelength. Since wavelength choices are governed by the ITU, there is little room for maneuver here: in the MNR domain, radars may operate with either of two wavelengths: 10 and 3 cm.
In reality, MNRs do not use parabolic dish antennas, but designs that mimic their essential characteristics; even so, they are constrained in much the same way and suffer many of the same issues, and their performance is degraded no less than their parabolic counterparts. To allow for this degradation, a reasonable ruleofthumb in common use for approximating an MNR antenna's beamwidth in degrees is similar to that shown above:
Dimension Beamwidth ≈ 70λ / d degrees
(d = dimension length in meters)
Very often, it is desirable to express the performance of an antenna system in terms of its gain with respect to an isotropic pattern. This requires knowledge of both the horizontal and the vertical beamwidths (BW) of an antenna; with MNR antenna designs, the latter may necessarily have to be deduced or assumed, since it is typically difficult  but not impossible  to characterize the vertical dimension. Fortunately, IMO regulations mandate the minimum vertical beamwidth that may be used, and manufacturer documentation may also yield specific data. Since the fanshaped radiation^{[3]} required of the MNR antenna is approximately rectangular, it may be expressed as follows:
Beam Area = BW_{Azimuth} * BW_{Elevation} degrees^{2}
Since the surface area of a sphere is 4π radians^{2}, or approximately 41253 square degrees, the gain of an antenna (with respect to the theoretical isotropic radiator, and once the beam area is known) may be expressed in decibel format as follows:
10 * Log_{10} 41253 : Beam Area (dBI),
where dBI = 'decibels compared to isotropic'
In reality, of course, the main beam does not contain the entire output from the transmitter; much of this is actually radiated in side lobes distributed widely around the main lobe  above, below, on both sides and behind the intended direction of radiation. Opinions differ marginally over how to account for this when estimating antenna gain, although in all instances their approach is to scale the mainbeam radiation to a smaller "surface", using a scalar substantially smaller than 41253 square degrees. One wellknown engineerscientist recommends 20000; another recommends using 25600; and a third, who has specialized in the design of marine navigation radars, recommends a value of 23500^{[4] }.
Beamwidth Measurement
The practical measurement of antenna beamwidth is discussed in more detail in Annex B; here, it is sufficient to say that an antenna's beamwidth may be defined either: as the arc subtended from the peak of the main beam to the first null in the antenna's radiation pattern; or as the arc subtended between the points on either side of the main beam where the measured power is exactly half of the power at the peak of the main beam.


